Power Rule of Derivatives

Power Rule of derivative is an essential formula in differential calculus. Now we shall prove this formula by definition or first principle.

Let us suppose that the function of the form y = f\left( x \right) = {x^n}, where n is any constant power.

First we take the increment or small change in the function.

\begin{gathered} y + \Delta y = {\left( {x + \Delta x} \right)^n} \\ \Rightarrow \Delta y = {\left( {x + \Delta x} \right)^n} - y \\ \end{gathered}

Putting the value of function y = {x^n} in the above equation, we get

 \Rightarrow \Delta y = {\left( {x + \Delta x} \right)^n} - {x^n}

Taking x common from the above equation, we get

 \Rightarrow \Delta y = {x^n}{\left( {1 + \frac{{\Delta x}}{x}} \right)^n} - {x^n}

Now taking common {x^n}, we get

 \Rightarrow \Delta y = {x^n}\left[ {{{\left( {1 + \frac{{\Delta x}}{x}} \right)}^n} - 1} \right]

Expanding the above expression using binomial series, we get

\begin{gathered} \Rightarrow \Delta y = {x^n}\left[ {1 + n\left( {\frac{{\Delta x}}{x}} \right) + \frac{{n\left( {n - 1} \right)}}{{2!}}{{\left( {\frac{{\Delta x}}{x}} \right)}^2} + \cdots - 1} \right] \\ \Rightarrow \Delta y = {x^n}\left( {\frac{{\Delta x}}{x}} \right)\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \Rightarrow \Delta y = {x^{n - 1}}\Delta x\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \end{gathered}

Dividing both sides by \Delta x, we get

\begin{gathered} \Rightarrow \frac{{\Delta y}}{{\Delta x}} = \frac{{{x^{n - 1}}\Delta x\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right]}}{{\Delta x}} \\ \Rightarrow \frac{{\Delta y}}{{\Delta x}} = {x^{n - 1}}\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \end{gathered}

Taking limit of both sides as \Delta x \to 0, we have

\begin{gathered} \Rightarrow \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} {x^{n - 1}}\left[ {n + \frac{{n\left( {n - 1} \right)}}{{2!}}\left( {\frac{{\Delta x}}{x}} \right) + \cdots } \right] \\ \Rightarrow \frac{{dy}}{{dx}} = {x^{n - 1}}\left[ {n + 0 + 0 + \cdots } \right] \\ \Rightarrow \frac{{dy}}{{dx}} = n{x^{n - 1}} \\ \Rightarrow \frac{d}{{dx}}{x^n} = n{x^{n - 1}} \\ \end{gathered}

This shows that the derivative of x power n is n{x^{n - 1}}.

Example: Find the derivative of y = f\left( x \right) = 7{x^3} + 2

We have the given function as

y = 7{x^3} + 2

Differentiation with respect to variable x, we get

\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {7{x^3} + 2} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = 7\frac{d}{{dx}}{x^3} + \frac{d}{{dx}}2 \\ \end{gathered}

Now using the power rule and constant function rule, we have

\frac{{dy}}{{dx}} = 21{x^2}